Does the continuous mode work properly when bracketing is off? According to the instructions, it ought to take three shots on continuous and then pause. If the continuous mode works correctly ordinarily, but not during bracketing, I'd guess there's a problem somewhere in the setting.
I note that WB bracketing does not work with Raw files. Nothing is said about AE and ADL bracketing, but you might want to double check whether that is an omission in the manual. Sorry I don't have a D5200 on which to experiment.
The D7100 appears to have a similar arrangement and says only that WB bracketing does not work in Raw, but also mentions nothing else. I'd double check first of all whether some step got omitted. As a second experiment, I'd see if bracketing works normally in single shot mode.
If you set bracketing in A mode you have to press the shutter 3 times before it takes bracketed exposure. This is good if you use a tripod, but if you are out walking for example, just take one raw and split later. Saves carrying a tripod.
It's not clear in the OP whether the object of the bracketing is HDR or just looking for the best exposure. If the former, I'd agree that splitting is probably easier and more accurate, since post processing is involved anyway. If you're just looking for the best exposure without post processing (for example to get off quick JPG's straight from the camera) it might still be worth doing, even though it's not hard to fix in post or even using the Raw processing in the camera.
Comments
I note that WB bracketing does not work with Raw files. Nothing is said about AE and ADL bracketing, but you might want to double check whether that is an omission in the manual. Sorry I don't have a D5200 on which to experiment.
The D7100 appears to have a similar arrangement and says only that WB bracketing does not work in Raw, but also mentions nothing else. I'd double check first of all whether some step got omitted. As a second experiment, I'd see if bracketing works normally in single shot mode.
Saves carrying a tripod.